### Notes from Toppers

**Mathematics: Circles Notes for JEE**

**Standard Equation of a Circle**

- Equation of a circle with center (h, k) and radius r: $${\bf (x - h)^2 + (y - k)^2 = r^2}$$

**Properties of Chords**

- Chords intersecting at right angles:
- Product of lengths of segments of one chord from the point of intersection is equal to the product of lengths of segments of the other chord.
- Tangents from an external point:
- Tangents drawn from an external point to a circle are equal in length.
- Lengths and positions of chords:
- The perpendicular from the center of the circle to a chord bisects the chord.

**Tangents**

- Equation of tangent at point (x₁, y₁) on circle with center (h, k): $${\bf (y - k) = \frac{y_1 - k}{x_1 - h} (x - h)}$$
- Conditions for tangency:
- Line has the same slope as the radius at the point of tangency.
- Line passes through the center of the circle.

**Angle Properties**

- Angle formed by two tangents from an external point: Equal
- Inscribed angles:
- Measure of an inscribed angle is half of the measure of the intercepted arc.
- Opposite angles in a cyclic quadrilateral are supplementary.

**Cyclic Quadrilaterals**

- Properties of cyclic quadrilaterals:
- Opposite angles are supplementary.
- Opposite sides multiply to give the same product.

- Condition for a quadrilateral to be cyclic:
- Sums of opposite angles are equal.

**Power of a Point**

- Power of a point (x₁, y₁) with respect to a circle with center (h, k) and radius r: $${\bf (x_1 - h)^2 + (y_1 - k)^2 - r^2}$$
- If the power of a point is positive, the point is outside the circle.
- If the power of a point is zero, the point lies on the circle.
- If the power of a point is negative, the point is inside the circle.

**Equations of Circles**

- General equation of a circle: $${\bf x^2 + y^2 + 2gx + 2fy + c = 0}$$
- Finding the center and radius of a circle from its equation

**Area and Circumference of Circles**

- Area: $${\bf A = \pi r^2\hspace{0.5}cm}$$
- Circumference: $${\bf C = 2\pi r\hspace{0.5}cm}$$

**Parametric Equations of Circles**

- Parametric equations of a circle with center (h, k) and radius r: $${\bf x = h + r\cos \theta, \hspace{0.5}cm y = k + r\sin \theta,\hspace{0.5}cm 0 \leq \theta \leq 2\pi}$$

**Intersection of Circles and Lines**

- Points of intersection:
- Solve the equations of the circle and line simultaneously to find the points of intersection.
- Tangency conditions:
- The distance from the center of the circle to the line is equal to the radius.
- The line is perpendicular to the radius at the point of tangency.

**Recommended Resources**

- NCERT Mathematics textbooks for Class 11 and Class 12.
- JEE Main and Advanced Previous Year Question Papers and Sample Papers.
- Reference books and study guides specifically for the JEE exam.
- Online resources and video tutorials related to Circle topics.